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In the algebraic group $G=\operatorname {PSO}(4,K)<\operatorname {PCGO}(4,K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative: $$ e=\left[ \begin{pmatrix} w & 0 & 0 & 0\\ 0 & w^{-1} & 0 & 0\\ 0 & 0 & w & 0\\ 0 & 0 & 0 & w^{-1} \\ \end{pmatrix} \right]$$ where $w$ is of order 4 in $K$.
The centralizer in $G$ is $C_G(e) =T_2.4$. The "4" is a Klein 4-group generated by $$ f_1=\left[ \begin{pmatrix} 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \\ \end{pmatrix} \right], f_2=\left[ \begin{pmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ \end{pmatrix} \right].$$ There is an $$ f=\left[ \begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \\ \end{pmatrix} \right]\in \operatorname {PCGO}(4,K) \setminus \operatorname {PSO}(4,K)$$ which also centralizes $e$.
Thus seems like $C_{\operatorname {PCGO}(4,K)}(e)= T_2.4.2$ and the $4.2$ is supposed to be a dihedral group of order 8. But the conjugate action of $f$ on $f_1$ and $f_2$ doesn't seem to give a dihedral group. What is going on here? Thank you!
Edit: $\operatorname{CGO}$ is the subgroup of the general linear group which fixes a non-degenerate quadratic form up to a scalar. $\operatorname {PCGO}$ is the projective group.